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Absolute value is a mathematician's way of judging numbers by their magnitude rather than their positive/negative value. It is the distance of the number from 0 on a number line.
Let \(x=70\) be one root of the equation \(2\lvert x-10\rvert=x+a,\) where \(a\) is a constant. Then what is the other root?
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Let \(S\) be the solution set of the equation \(\lvert 2x-1 \rvert + 3\lvert x-29\rvert=5x-88.\) Then what is \(S?\)
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How many ordered pairs \((x, y)\) are there such that \(x\) is a nonzero integer that satisfies \(-8\leq x \leq 8\), \(y\) is a nonzero integer that satisfies \(-2\leq y \leq 2\), and \[\lvert x+y \rvert = \lvert x \rvert + \lvert y \rvert?\]
Details and assumptions
For an ordered pair of integers \((a,b)\), the order of the integers matter. The ordered pair \((1, 2)\) is different from the ordered pair \((2,1) \).
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How many ordered triples of integers \( (a, b, c) \) subject to \( -20 < a < b < c < 20,\) are there, such that
\[ \left| a + b + c \right | = |a| + |b| + |c|? \]
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If \(20 < a \leq 21\), what is the value of \[ \bigg| 2-a- \big| 6- \lvert a-20 \rvert \big| \bigg| ?\]
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